Equivalence principle in chameleon models

Most theories that predict time and/or space variation of fundamental constants also predict violations of the weak equivalence principle (WEP). In 2004 Khoury and Weltman [1] proposed the so called chameleon field arguing that it could help avoiding experimental bounds on the WEP while having a nontrivial cosmological impact. In this paper we revisit the extent to which these expectations continue to hold as we enter the regime of high precision tests. The basis of the study is the development of a new method for computing the force between two massive bodies induced by the chameleon field which takes into account the influence on the field by both, the large and the test bodies. We confirm that in the thin shell regime the force does depend nontrivially on the test body’s composition, even when the chameleon coupling constants ${\beta }^i=\beta$ are universal. We also propose a simple criterion based on energy minimization, that we use to determine which of the approximations used in computing the scalar field in a two body problem is better in each specific regime. As an application of our analysis we then compare the resulting differential acceleration of two test bodies with the corresponding bounds obtained from Eötvös type experiments. We consider two setups: (1) an Earth based experiment where the test bodies are made of Be and Al; (2) the Lunar Laser Ranging experiment. We find that for some choices of the free parameters of the chameleon model the predictions of the Eötvös parameter are larger than some of the previous estimates. As a consequence, we put new constrains on these free parameters. Our conclusions strongly suggest that the properties of immunity from experimental tests of the WEP, usually attributed to the chameleon and related models, should be carefully reconsidered. An important result of our analysis is that our approach leads to new constraints on the parameter space of the chameleon models.

Figure 1

Effective potential inside and outside the large body. The blue line shows the potential $V_{\mathrm{eff}}(\phi )$ for the one body problem, the red line shows $V_{\mathrm{eff}}(\phi )$ for $\rho ={\rho }_{\mathrm{out}}$ in Eq. (7), the green line shows $V_{\mathrm{eff}}(\phi )$ for $\rho ={\rho }_{\mathrm{in}}$ in Eq. (7) and the light blue line shows the quadratic approximation Left: The thick shell case; Right: The thin shell case.

Figure 2

Two body problem.

Figure 3

Coordinates transformation.

Figure 4

The chameleon field $\phi$ as a function of coordinate $z$ inside the bodies and in their outskirts ($z$ is the distance to the center of the large body in cm, then $\phi$ is measured in $\mathrm{c}{\mathrm{m}}^{-1}$). The large body consists of a sphere of radius $R_1=1550\mathrm{cm}$ and ${\mu }_1=197.2244{\mathrm{cm}}^{-1}$. This body mimics roughly the actual hill where the experiment of the Eöt-Wash group takes place and which produces the desired combined effects (gravitational and “fifth” force if any). The test body consists of a 10 gr sphere of aluminium and has a radius $R_2=0.96\mathrm{cm}$ and ${\mu }_2=78.1940{\mathrm{cm}}^{-1}$. In both figures $n=1=\beta =1$ and $M=2.4\mathrm{meV}$ and ${\rho }_{\mathrm{out}}={\rho }_{\mathrm{atm}}={10}^{-3}\mathrm{g}{\mathrm{cm}}^{-3}$ Red line: our approach, blue line: standard approach [1].

Figure 5

Relative difference of energies $\left(\frac{U_{\text{standard}}-U_{\text{our approach}}}{U_{\text{standard}}}\right)$ computed for the Eöt-Wash experiment. The density of the environment surrounding both the test body (Al) and the source (the mountain) is assumed to be the same. Left: $M=2.4\mathrm{meV}$ and ${\rho }_{\mathrm{out}}={10}^{-7}\mathrm{g}{\mathrm{cm}}^{-3}$ for the density of the vacuum chamber. The thin shell condition for the mountain is always satisfied, while for $\beta$ lower than the value indicated by the vertical dotted line the test body does not have a thin shell. Right: $M=10\mathrm{eV}$ and ${\rho }_{\mathrm{out}}={10}^{-3}\mathrm{g}{\mathrm{cm}}^{-3}$ for the density of the atmosphere. For values of $\beta$ lower than those indicated by the vertical dotted lines the bodies have no thin shell.

Figure 6

Comparison of the energy computed for the LLR experiment using our approach (red) and the standard one (blue) taking $M=10\mathrm{eV}$. The test body is assumed to be the Earth and the environment corresponds to the interstellar medium with density ${\rho }_{\mathrm{out}}={10}^{-24}\mathrm{g}{\mathrm{cm}}^{-3}$. The vertical dotted lines show the thin shell condition for Earth and Sun (for values of $\beta$ lower than those indicated by the vertical dotted lines the bodies have no thin shell).

Figure 7

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) for different positive values of $n$ and $M=2.4\times {10}^{-3}\mathrm{eV}$. The density of the environment ${\rho }_{\mathrm{out}}$ is assumed to be equal to: Panel (a) the vacuum’s density inside the vacuum chamber ${\rho }_{\mathrm{out}}={10}^{-7}\mathrm{g}{\mathrm{cm}}^{-3}$; Panel (b) the Earth’s atmosphere density ${\rho }_{\mathrm{out}}={10}^{-3}\mathrm{g}{\mathrm{cm}}^{-3}$. The predictions computed by Khoury and Weltman [1] are also included and labeled as the standard approach.

Figure 8

The two body problem including the metal encasing of the vacuum chamber.

Figure 9

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) including the metal encasing of the vacuum chamber as depicted in Fig. 8 for different positive values of $n$. Here we assume that both the density of the environment surrounding the Earth and inside the vacuum chamber are ${\rho }_{\mathrm{out}}={10}^{-7}\mathrm{g}{\mathrm{cm}}^{-3}$. Also shown are the predictions computed by Khoury and Weltman [1] with the effect of the metal encasing modeled by multiplying their estimates of $\eta$ by the factor $\mathrm{sech}(2m_{\text{shell}}d)$.

Figure 10

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) including the metal encasing of the vacuum chamber as shown in Fig. 8 for different positive values of $n$. The density of the environment surrounding the Earth ${\rho }_{\mathrm{out}}$ is assumed to be equal the Earth’s atmosphere density ${\rho }_{\mathrm{out}}={10}^{-3}\mathrm{g}{\mathrm{cm}}^{-3}$; the density of the environment inside the vacuum chamber is assumed to be ${\rho }_{\mathrm{vac}}={10}^{-7}\mathrm{g}{\mathrm{cm}}^{-3}$; Left: $M=2.4\mathrm{meV}$ (cosmological chameleon); Right: $M=10\mathrm{eV}$.

Figure 11

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) for the LLR experiment as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) for different positive values of $n$. All bodies are surrounded by the interstellar medium. $M=2.4\mathrm{meV}$ (cosmological chameleon). In all cases we compare the results of the calculations performed in this paper with the predictions computed by Khoury and Weltman [1].

Figure 12

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) for the LLR experiment as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) for different positive values of $n$. All bodies are surrounded by the interstellar medium. $M=10\mathrm{eV}$. The vertical lines show the values of $\beta$ below which the thin shell condition is no longer satisfied for the Earth, Moon and Sun. For all values computed in this plot, the energy criterion developed in Sec. 3 indicates that our approximation to the effective potential is better than the one used by the standard approach.

Figure 13

The shaded region in the figure shows the values in the $n\text{−}\beta$ plane where our approach has significant differences with the standard approach in the value of $\eta$ for the LLR experiment with $M=10\mathrm{eV}$.

Figure 14

Bounds in the $n\text{−}\beta$ plane obtained in this paper (in red) using the torsion balance experiment with combined constraints compiled by Sakstein and Burrage. Here $M=2.4\mathrm{meV}$.

Figure 15

The Eötvös parameter $\eta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) as a function of the parameter $\beta$ (in $\mathrm{lo}{\mathrm{g}}_{10}$ scale) including the metal encasing of the vacuum chamber as shown in Fig. 8 for different positive values of $n$ and $M=2.4\mathrm{meV}$. Here we assume that both the density of the environment surrounding the Earth (and of course, the hill) and inside the vacuum chamber are ${\rho }_{\mathrm{out}}={10}^{-7}\mathrm{g}{\mathrm{cm}}^{-3}$. We compare our predictions considering the metal encasing in the calculations with those of ours too, but where the effect of the metal encasing is modeled multiplying a factor $\mathrm{sech}(2m_{\text{shell}}d)$ to the predictions.